mhaviTüal½ysMNg;sIuvil                                                         viTüasßanCatiBhubec©keTskm<úCa


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Department of Civil Engineering                                                                NPIC



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mhaviTüal½ysMNg;sIuvil                                                   viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                                  NPIC



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mhaviTüal½ysMNg;sIuvil                                                   viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                                NPIC



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mhaviTüal½ysMNg;sIuvil                                                    viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                                 NPIC



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mhaviTüal½ysMNg;sIuvil                                                                viTüasßanCatiBhubec©keTskm<úCa




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Department of Civil Engineering                                                              NPIC



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mhaviTüal½ysMNg;sIuvil                                                  viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                      NPIC


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mhaviTüal½ysMNg;sIuvil                                                            viTüasßanCatiBhubec©keTskm<úCa




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Department of Civil Engineering                                                                    NPIC


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mhaviTüal½ysMNg;sIuvil                                                 viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                              NPIC


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mhaviTüal½ysMNg;sIuvil                                                              viTüasßanCatiBhubec©keTskm<úCa




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Department of Civil Engineering                                                         NPIC



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mhaviTüal½ysMNg;sIuvil                                                viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                           NPIC



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15. beam analysis using the stiffness method

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15. beam analysis using the stiffness method

  1. 1. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa !%> karviPaKFñwmedayeRbIviFIPaBrwgRkaj (Beam analysis using the stiffness method) emeronenHnwgbgðajBIeKalKMnitEdlerobrab;enAkñúgemeronmunedIm,IviPaKFñwm. eyIgeXIjfa enA eBlEdleKbegáItm:aRTIsPaBrwgRkajsRmab;Ggát; nigm:aRTIsbMElg dMeNIrkarsRmab;karGnuvtþmanlkçN³ RsedogKñanwgkarGnuvtþsRmab; truss. emeronenHnwgbgðajBIkrNIBiesssRmab;sRmutDIepr:g;Esül nigsItuNðPaB. !%>!> kMNt;cMNaMbzm (preliminary remarks) munnwgeyIgbgðajBIrebobEdlviFIPaBrwgRkajGnuvtþeTAelIFñwm dMbUgeyIgnwgerobrab;BIeKalKMnit bzm nigniymn½yEdlerobrab;eTAnwgGgát;TaMgenH. karkMNt;sMKal; node nigGgát;³ edIm,IGnuvtþviFIPaBrwgRkajeTAelIFñwm dMbUgeyIgRtUvEtkMNt;BIrebobbMEbk FñwmCabgÁúMFatuGnnþtUc (finite element). CaTUeTA FatunImYy²minRtUvTak;TgnwgbnÞúkeT ehIyRtUvmanmux kat;efr. sRmab;mUlehtuenH node rbs;FatuRtUvEtsßitenARtg;TMr b¤Rtg;cMNucEdlGgát;P¢ab;Kña Rtg;bnÞúk Gnuvtþ Rtg;RkLaépÞmuxkat;pøas;bþÚrPøam² b¤Rtg;cMNucEdleKRtUvkMNt;bMlas;TImMu. ]TahrN_ eKmanFñwm dUcbgðajenAkñúgrUbTI 15-1a. edayeRbIviFIdUcKñasRmab;karviPaK truss eKRtUvkMNt; node bYnCaelxeday dak;kñúgrgVg;. dUcKña cMNaMfa cugCit nigcugq¶ayrbs;Ggát;nImYy²RtUv)ankMNt;sMKal;edaysBaØaRBYjtam beNþayGgát;nImYy². kUGredaenskl nigkUGredaen³ eyIgnwgkMNt;RbB½n§kUGredaenskledayeRbIG½kS x, y, z EdleKalrbs; vasßitenARtg; node EdleFVIeGaykUGredaenrbs;cMNucdéTeTotenAelIFñwmviC¢man ¬rUbTI 15-1a¦. kUGr- Beam analysis using the stiffness method T.Chhay -501
  2. 2. Department of Civil Engineering NPIC edaentMbn; nigkUGredaensRmab;Ggát; x' , y' , z' maneKalrbs;vasßitenAcugCitrbs;Ggát;nImYy² G½kS viC¢man x' manTisedAeTArkcugq¶ay. rUbTI 15-1b bgðajkUGredaenTaMgenHsRmab;Ggát;elx @. enAkñúg krNITaMgBIr eyIg)aneRbIRbB½n§kUGredaenédsþaM dUcenHRbsinebIeyIgbgVilédsþaMBIG½kS x b¤ x' eTAG½kS y b¤ y ' eyIgnwgTTYl)anG½kS z b¤ z ' viC¢man EdlmanTisedAecjBIRkdasesovePA. cMNaMfa sRmab;Ggát;Fñwm nImYy²G½kS x nig x' nwgsßitenAelIG½kSEtmYy ehIykUGredaenskl nigkUGredaensRmab;Ggát;TaMgGs;suT§ EtRsbKña. dUcenH ¬mindUckrNIsRmab; truss¦ enATIenH eyIgnwgmincaM)ac;begáItm:aRTIsbMElgrvagRbB½n§ kUGredaenTaMgenH. PaBminkMNt;sIueNm:aTic³ eRkayeBlGgát; nig node RtUv)ankMNt; nigeRkayeBlRbB½n§kUGredaenskl RtUv)anbegáIt eKGackMNt; degree of freedom nigPaBminkMNt;sIueNm:aTicsRmab;Fñwm. RbsinebIeyIg KitTaMgT§iBlm:Um:g;Bt; nigkmøaMgkat; enaH node nImYy²enAelIFñwmGaclman degree of freedom BIreBal KW bMlas;TItamTisQr nigmMurgVil. dUcenAkñúgkrNIrbs; truss eKGackMNt;bMlas;TITaMgenHedayelxkUd. eyIgeRbIelxkUdtUcbMputedIm,IeFVIkMNt;sMKal;bMlas;TIEdlCaGBaØat ¬degree of freedom Edlminmankar Tb;¦ ehIyeyIgeRbIelxFMbMputedIm,IeFVIkMNt;sMKal;bMlas;TIEdleyIgsÁal; ¬degree of freedom Edlman karTb;¦. rMlwkfa mUlehtukñúgkareRCIserIsviFIénkareFVIkMNt;sMKal; eyIgRtUveFVIkarEbgEckm:aRTIsPaBrwg RkajsRmab;rcnasm½<n§eGay)anRsYl dUcenHeyIgGacKNnabM;las;TIEdlCaGBaØattamrebobedaypÞal; bMput. edIm,Ibgðaj]TahrN_kñúgkardak;elxkUd eyIgnwgBicarNaFñwmCab;mþgeTotenAkñúgrUbTI 15-1a. enA TIenH FñwmCarcnasm<½n§minkMNt;edaysIueNm:aTicdWeRkTIbYn. eKman degree of freedom cMnYnR)aMbI Edl elxkUd ! dl;elxkUd $ tMNageGaybMlas;TIEdlCaGBaØat ehIyelxkUd % dl;elxkUd * tMNageGay bMlas;TIEdleKsÁal; ¬EdlenAkñúgkrNIenHvasuT§EtmantémøesμIsUnü¦. ]TahrN_epSgeTot eKGacbMEbk FñwmenAkñúgrUb TI 15-2a CabIGgát; nigbYn node. cMNaMfa snøak;enARtg; node elx # dabdUcGgát;elx @ karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -502
  3. 3. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa nigelx # b:uEnþmMurgVilenARtg;cugGgát;nImYy²mantémøxusKña. enATIenH eKman degree of freedom cMnYn R)aMbYn EdlenAkñúgenaH degree of freedom R)aMCaGBaØat ¬dUcbgðajenAkñúgrUbTI 15-2b¦ ehIy degree of freedom cMnYnbYnRtUv)ansÁal; ¬vaTaMgGs;suT§EtesμIsUnü¦. cugeRkay BicarNakarrGil ¬slider mechanism¦ EdleRbIenAelIFñwmenAkñúgrUbTI 15-3a. enATIenH PaBdabrbs;FñwmRtUv)anbgðajenAkñúgrUbTI 15-3b dUc enHeKmanbgÁúMPaBdabEdlCaGBaØatcMnUnR)aMEdlRtUv)ankMNt;elxerogCamYynwgelxkUdtUcbMput. FñwmCa eRKOgbgÁúMminkMNt;edaysIeu Nm:aTicdWeRkTIR)aM. karbegáItviFIPaBrwgRkajsRmab;FñwmmanlkçN³RsedogeTAnwgdMeNIrkarkñúgkarbegáItPaBrwgRkaj EdleRbIsRmab; truss. dMbUg eyIgRtUvbegáItm:aRTIsPaBrwgRkajsRmab;Ggát;nImYy² ehIybnÞab;mkpÁúMm:aRTIs TaMgenHedIm,IbegáItm:aRTIsPaBrwgRkajsRmab;Fñwm b¤m:aRTIsPaBrwgRkajsRmab;rcnasm½<n§. edayeRbIsmIkar m:aRTIssRmab;rcnasm<½n§ eyIgGacbnþkarKNnaedIm,IkMNt;bM;las;TIEdlCaGBaØatenARtg; node ehIyBIkar kMNt;enH eyIgGackMNt;RbtikmμenAelIFñwm nigkmøaMgkat; nigm:Um:g;enARtg; node. !%>@> m:aRTIsPaBrwgRkajsMrab;Ggát;Fñwm (Beam-member stiffness matrix) enAkñúgkfaxNÐenH eyIgnwgbegáItm:aRTIsPaBrwgRkajsRmab;FñwmEdlmanRkLaépÞmuxkat;efr nig manRbB½n§kUGredaentMbn; x' , y' , z' ¬rUbTI 15-4¦. eKRtUvdak;eKalrbs;kUGredaenenARtg;cugCit N ehIyG½kSviC¢manrbs; x' manTisedAeTArkcugq¶ay F . eKmankmøaMgRbtikmμBIrenARtg;cugnImYy²rbs;Ggát; EdlmankmøaMgkat; q N nig q F nigm:Um:g;Bt; q N nig q F . bnÞúkTaMgenHeFVIGMeBItamTisedAkUGredaenviC¢- z' y' z' z' man. m:Um:g; q N nig q F viC¢manvilRcasTisRTnicnaLika tamviFanédsþaMviucT½rm:Um:g;manTistambeNþay z' z' G½kSviC¢man z' EdlmanTisecjBIRkdasesovePA. Beam analysis using the stiffness method T.Chhay -503
  4. 4. Department of Civil Engineering NPIC bMlas;TIlIenEG‘r nigbMlas;TImMuEdlpSMCamYynwgbnÞúkTaMgenHk¾GnuvtþtamkarkMNt;sBaØaviC¢mandUc Kña. enATIenH eyIgnwgdak;bMlas;TITaMgenHdac;edayELkBIKña ehIybnÞab;mkkMNt;bnÞúkEdlmanGMeBIenAelI Ggát;EdlbNþalBIbMlas;TInImYy². bMlas;TI y' ³ enAeBleKeFVIeGaymanbMlas;TIviC¢man d N xN³eBlEdleKGackarBarbMlas;TIEdlGac y' ekItman eKGacbegáItkmøaMgkat; nigm:Um:g;Bt;RtUv)anbgðajenAkñúgrUbTI 15-5a. m:Um:g;RtUv)anbegáItenAkñúg kfaxNÐ 11-2 dUcsmIkar 11-5. dUcKña enAeBleKeFVIeGayman d F eyIgTTYl)ankmøaMgkat; nigm:Um:g;dUc y' bgðajenAkñúgrUbTI 15-5b. mMurgVil z' ³ RbsinebIeKeFVIeGaymanmMurgVil d N xN³bMlas;TIEdlGacekItmanTaMgGs;RtUv)ankarBar z' kmøaMgkat; nigm:Um:g;EdlcaM)ac;edIm,IeFVIeGaymankMhUcRTg;RTayRtUv)anbgðajenAkñúgrUbTI 15-6a. m:Um:g; RtUv)anbegáItenAkñúgkfaxNÐ 11-2 dUcsmIkar 11-1 nig 11-2. dUcKña enAeBleKeFVIeGayekItman d F bnÞúk z' pÁÜbRtUv)anbgðajenAkñúgrUbTI 15-6b. karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -504
  5. 5. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa edayviFItRmYtpl RbsinebIeKbUklT§plenAkñúgrUbTI 15-5 nig 15-6 bBa©ÚlKña eyIgGacsresr TMnak;TMngrvagbnÞúk nigbMlas;TITaMgbYnsRmab;Ggát;enAkñúgTRmg;m:aRTIsCa N y' N z' F y' Fz' ⎡ 12 EI 6 EI 12 EI 6 EI ⎤ ⎢ 3 2 − 3 ⎥ ⎡q N y ' ⎤ ⎢ 6L L L L2 ⎥ d ⎢ q ⎥ ⎢ EI 4 EI 6 EI 2 EI ⎥ ⎡ N y ' ⎤ − 2 ⎢ ⎥ ⎢ N z ' ⎥ = ⎢ L2 L L L ⎥⎢d Nz' ⎥ (15-1) ⎢ q Fy ' ⎥ ⎢ 12 EI 6 EI 12 EI 6 EI ⎥ ⎢ d ⎥ ⎢ ⎥ ⎢− 3 − 2 − 2 ⎥ ⎢ Fy ' ⎥ ⎣ q Fz ' ⎥ ⎢ L ⎢ ⎦ L L3 L ⎥⎢ d F ⎥ ⎢ 6 EI 2 EI 6 EI 4 EI ⎥ ⎣ z ' ⎦ − 2 ⎢ L2 ⎣ L L L ⎥ ⎦ eyIgk¾GacsresrsmIkarTaMgenHkñúgTRmg;kat;dUcxageRkam q = kd (15-2) m:aRTIssIuemRTI k enAkñúgsmIkar15-1 Cam:aRTIsPaBrwgRkajsRmab;Ggát;. emKuNT§iBl kij TaMg 16 EdlpSMCam:aRTIs k )anKitbBa©ÚlbMlas;TIEdlbNþalBIkmøaMgkat; nigm:Um:g;Bt;rbs;Ggát;. tamrUbviTüa emKuNTaMgenHCabnÞúkenAelIGgát; enAeBlGgát;rgbM;las;TIÉktþa. ]TahrN_ RbsinebI d N = 1 ¬rUbTI 15 y' -5a¦ xN³EdlbMlas;TIdéTeTotesμIsUnü Ggát;nwgrgEtbnÞúkbYnEdlbgðajenAkñúgCYrQrTImYyrbs;m:aRTIs k . tamrebobdUcKña CYrQrdéTeTotrbs;m:aRTIs k CabnÞúkenAelIGgát;sRmab;bMlas;TIÉktþaEdlkMNt; edayelxkUd degree of freedom RtUv)anbgðajBIxagelICYrQr. BIkarbegáItenH eKRtUvbMeBjlkçxNÐ lMnwg niglkçxNÐRtUvKña. dUcKña eyIgRtUvcMNaMfam:aRTIsenHdUcKñasRmab;kUGredaenskl nigkUGredaentMbn; edaysarG½kS x' , y' , z' RsbnwgG½kS x, y, z dUcenHeKminRtUvkarm:aRTIsbMElgrvagkUGredaenTaMgBIreT. !%>#> m:aRTIsPaBrwgRkajsMrab;eRKOgbgÁúMFñwm (Beam-structure stiffness matrix) eRkayeBlrkma:RTIsPaBrwgRkajsRmab;Ggát; eyIgRtUvpÁúMvaeTAkñúgm:aRTIsPaBrwgRkajsRmab; eRKOgbgÁúM K . dMeNIrkarenHGaRs½ynwgkarsÁal;TItaMgrbs;FatunImYy²enAkñúgm:aRTIsPaBrwgRkajsRmab; Ggát;CadMbUg. enATIenH CYredk nigCYrQrrbs;m:aRTIs k nImYy² ¬smIkar 15-1¦ RtUv)ankMNt;edayelx kUdBIrRtg;cugCitrbs;Ggát; ¬ N y' , N z' ¦ EdlbnþedayelxkUdBIrRtg;cugq¶ay ¬ Fy' , Fz' ¦. dUcenH enA eBlpÁúMm:aRTIs eKRtUvdak;FatunmYy²enATItaMgdUcKñarbs;m:aRTIs K . tamviFIenH K RtUvmanlMdan;esμInwgelx I kUdFMbMputEdlRtUv)ankMNt;eTAelIFñwm edaysarvaCaelxsrubén degree of freedom. dUcKña enATItaMg EdlGgát;CaeRcInP¢ab;Kñanwg node, emKuNT§iBlPaBrwgRkajsRmab;Ggát; (member stiffness influence Beam analysis using the stiffness method T.Chhay -505
  6. 6. Department of Civil Engineering NPIC coefficients) nwgmanTItaMgdUcKñaenAkñúgm:aRTIs K dUcenHvaRtUv)anbUkbBa©ÚlKñatamlkçN³nBVnþedIm,IkMNt; emKuNT§iBlPaBrwgRkajén node sRmab;rcnasm<½n§. eKcaM)ac;RtUveFVIEbbenH edaysaremKuNnImYy²Ca ersIusþg;rbs; node énrcnasm<½n§enAkñúgTisedA ¬ y' b¤ z' ¦ enAeBlekItmanbMlas;TIÉktþa ¬ y' b¤ z' ¦ enA Rtg; node dEdl b¤Rtg; node déT. ]TahrN_ K 23 CabnÞúktamTis nigenARtg;TItaMgénelxkUd@ enAeBl bMlas;TIÉktþaekItmantamTis nigenARtg;TItaMgénelxkUd #. !%>$> karGnuvtþénviFIPaBrwgRkajsMrab;karviPaKFñwm (Application of the stiffness method for bema analysis) eRkayeBleKkMNt;m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§rYcehIy eKGacP¢ab;TMnak;TMngrvagbnÞúk enARtg; node rbs;FñwmeTAnwgbMlas;TIedayeRbIsmIkarPaBrwgRkajsRmab;rcnasm<½n§ Q = KD enATIenH Q nig D Cam:aRTIsCYrQrEdltMNageGaybnÞúk nigbMlas;TIEdlCaGBaØat nigEdlsÁal;. karbMEbkm:aRTIsPaBrwgRkajeGayeTACaFatuénbnÞúk nigbMlas;TIEdlsÁal; nigGBaØat eyIgTTYl)an ⎡Qk ⎤ ⎡ K11 K12 ⎤ ⎡ Du ⎤ ⎢Q ⎥ = ⎢ K ⎥⎢ ⎥ ⎣ u ⎦ ⎣ 21 K 22 ⎦ ⎣ Dk ⎦ enAeBleyIgBnøatm:aRTIsxagelI eyIgnwgTTYl)ansmIkarBIrKW Qk = K11 Du + K12 Dk (15-3) Qu = K 21 Du + K 22 Dk (15-4) eKGackMNt;bMlas;TIEdlCaGBaØatBIsmIkarTImYy. edayeRbIsmIkarTaMgenH eyIgGacKNnakmøaMgRbtikmμ TMr Qu BIsmIkarTIBIr. bnÞúkEdlenAkNþal³ sRmab;karGnuvtþ eKcaM)ac;eGayFaturbs;FñwmminrgbnÞúktambeNþayFñwm edaysar m:aRTIsPaBrwgRkajsRmab;Ggát;RtUv)anbegáItsRmab;bnÞúkEdlGnuvtþEtenAxagcugrbs;va ¬emIlrUbTI 15- 4¦. b:uEnþ eBlxøHFñwmRtUvRTbnÞúkBRgay dUcenHeKRtUvkarEktRmUvlkçxNÐenHedIm,IGnuvtþkarviPaKeday m:aRTIs. edIm,IedaHRsaykrNIenH eyIgnwgeRbIeKalkarN_tRmYtpltamrebobRsedogKñaeTAnwgGIVEdl)an eRbIsRmab; truss Edl)anerobrab;enAkñúgkfaxNÐ 14-8. edIm,IbgðajBIkarGnuvtþrbs;va BicarNaFaturbs; FñwmEdlmanRbEvg L dUcbgðajenAkñúgrUbTI 15-7a EdlrgbnÞúkBRgayesμI w . dMbUg eyIgnwgGnuvtþkmøaMg karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -506
  7. 7. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa bgáb;cug nigm:Um:g;bgáb;cugeTAelIGgát; EdlvaRtUv)aneKeRbIenAkñúgviFIPaBrwgRkaj ¬rUbTI 15-7b¦. eyIgnwg sresrbnÞúkTaMgenHCam:aRTIsCYrQr q0 . bnÞab;mkeTot eyIgGnuvtþbnÞúkBRgayesμI nigkmøaMgRbtikmμ q0 rbs;va ¬rUbTI 15-7c¦. eKkMNt;bnÞúkCak;EsþgenAkñúgFñwmedayeFVIplbUkeTAelIlT§plTaMgBIr. kmøaMg Rbtikmμbgáb;cugsRmab;krNIbnÞúkdéTeTotRtUv)aneGayenAkñúgemeronTI11. bEnßmelIkaredaHRsaycMeNaT EdlBak;B½n§nwgbnÞúkxagdUckrNIenH eyIgk¾GaceRbIviFIenHedIm,IedaHRsaycMeNaTEdlBak;B½n§nwgbERmbRmYl sItuNðPaB b¤kMhuskñúgplitkmμ. kmøaMgkñúgGgát;³ eKGackMNt;kmøaMgkat; nigm:Um:g;Bt;enARtg;cugrbs;Ggát;FñwmnImYy²edayeRbIsmIkar 15-2 ehIybUkbEnßmkmøaMgRbtikmμbgáb;cug q0 RbsinebIGgát;rgbnÞúkenAkNþal. eyIgman q = kd + q0 (15-5) RbsinebIlT§plGviC¢man vabgðajfabnÞúkeFVIGMeBIpÞúynwgTisedAEdl)anbgðajenAkñúgrUbTI 15-4. dMeNIrkarkñúgkarviPaK (Procedure for analysis) viFIxageRkampþl;nUvmeFüa)ayedIm,IkMNt;bMlas;TI RbtikmμTMr kmøaMgkñúgrbs;Ggát; b¤FatuGnnþtUc rbs;FñwmkMNt;edaysþaTic b¤FñwmminkMNt;edaysrsþaTic. kareFVIkMNt;sMKal;³ bMEbkFñwmCaFatuGnnþtUc ehIykMNt;elxerogeGayFatunImYy² nig node nImYy²rbs;va. eRbIelx Beam analysis using the stiffness method T.Chhay -507
  8. 8. Department of Civil Engineering NPIC EdlsresrkñúgrgVg;sRmab; node nigelxkñúgkaersRmab;Ggát; b¤Fatu. CaTUeTAGgát; b¤FatusßitenA cenøaHTMr b¤enAcenøaHbnÞúkcMcMNuc b¤enAcenøaHtMN b¤enAcenøaHcMNucEdleKRtUvkarkMNt;kmøaMgkñúg b¤bMlas;TI. kMNt;cugCit nigcugq¶ayrbs;Ggát;edaysBaØaRBYjEdlsßitenAelIGgát; ehIymanTisedAeTArkcug q¶ay. enARtg; node nImYy² kMNt;elxkUdtamTis y nigtamTis z . enARKb;krNITaMgGs;eRbIelxkUdtUc bMputedIm,IsMKal; degree of freedom EdlminmankarTb; ehIybnþedayelxbnþbnÞab; b¤elxEdl FMbMputedIm,IsMKal; degree of freedom EdlmankarTb;. begáItm:aRTIsbMlas;TIEdlsÁal; Dk nigm:aRTIsbnÞúkEdlsÁal; Qk . KitbBa©Úlm:aRTiscRmas;én bnÞúkbgáb;cug RbsinebIGgát;rgbnÞúk. m:aRTIsPaBrwgRkajsRmab;rcnasm½<n§³ GnuvtþsmIkar 15-1 edIm,IkMNt;m:aRTIsPaBrwgRkajsRmab;FatunImYy²edayeRbIkUGredaenskl. eRkayeBlkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát; ehIyCYredk nigCYrQrRtUv)ankMNt;CamYynwg elxkUdsmRsb pÁúMm:aRTIsedIm,ITTYl)anm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§ K . sRmab;kar epÞógpÞat;edayEpñk m:aRTIsPaBrwgRkajsRmab;Ggát; nigm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§RtUv EtCam:aRTIssIuemRTI. bMlas;TI nigbnÞúk³ bMEbksmIkarPaBrwgRkajsRmab;rcnasm<½n§CaRkum ehIyeFVIRbmaNviFIKuNm:aRTIsedIm,IkMNt; bMlas;TIEdlCaGBaØat Du nigkmøaMgRbtikmμTMr Qu . eKGackMNt;kmøaMgkat; nigm:Um:g; q enARtg;cugrbs;Ggát;FñwmnImYy²BIsmIkar 15-5 edayKitbBa©Úl TaMgbnÞúkbgáb;cug. ]TahrN_ 15-1³ kMNt;kmøaMgRbtikmμenARtg;TMrrbs;FñwmEdlbgðajenAkñúgrUbTI 15-8a. EI CacMnYnefr. karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -508
  9. 9. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ kareFVIkMNt;smÁal;³ FñwmmanFatuBIr nig node bI EdlRtUv)ankMNt;dUcbgðajenAkñúgrUbTI 15-8b. eKeRbI elxkUd 1 dl;elxkUd$ sRmab;kMNt; degree of freedom EdlminrgkarTb;. m:aRTIsbnÞúkEdlsÁal; nigbMlas;TIEdlsÁal;KW ⎡ 0 ⎤1 ⎢ − 5⎥ 2 ⎡0 ⎤ 5 Qk = ⎢ ⎥ Dk = ⎢ ⎥ ⎢ 0 ⎥3 ⎣0 ⎦ 6 ⎢ ⎥ ⎣ 0 ⎦4 m:aRTIsPaBrwgRkajsRmab;Ggát;³ eKkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;TaMgBIrBIsmIkar 15-1. eyIg RtUvcMNaMBIrebobbegáItelxkUdsRmab;CYredk nigCYrQr. 6 4 5 3 5 3 2 1 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 6 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 5 ⎢ − 1.5 1 ⎥ 4 ⎢ − 1.5 1 ⎥ 3 k1 = EI ⎢ 1.5 2 ⎥ k 2 = EI ⎢ 1.5 2 ⎥ ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 5 ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 2 ⎢ ⎥ ⎢ ⎥ ⎣ 1.5 1 − 1.5 2 ⎦ 3 ⎣ 1.5 1 − 1.5 2 ⎦ 1 bMlas;TI nigbnÞúk³ eyIgGacpÁúMFatuTaMgenHeTAkñúgm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§. ]TahrN_ Fatu K11 = 0 + 2 = 2, K 55 = 1.5 + 1.5 = 3 .l. dUcenH Q = KD 1 2 3 4 5 6 ⎡0⎤ ⎡ 2 − 1.5 1 0 1.5 0 ⎤ ⎡ D1 ⎤ ⎢ − 5⎥ ⎢− 1.5 1.5 − 1.5 0 0 ⎥ ⎢ D2 ⎥ − 1.5 ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ 0 ⎥ = EI ⎢ 1 − 1.5 4 1 0 1.5 ⎥ ⎢ D3 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢0⎥ ⎢ 0 0 1 2 − 1.5 1.5 ⎥ ⎢ D4 ⎥ ⎢Q5 ⎥ ⎢ 1.5 − 1.5 0 − 1.5 3 − 1.5⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q6 ⎥ ⎣ ⎦ ⎢ 0 ⎣ 0 1.5 1.5 − 1.5 1.5 ⎥ ⎢ 0 ⎥ ⎦⎣ ⎦ m:aRTIsRtUv)anbMEbkCaRkumdUcbgðaj. edayedaHRsayplKuNsRmab;CYredkbYndMbUg eyIgTTYl)an Beam analysis using the stiffness method T.Chhay -509
  10. 10. Department of Civil Engineering NPIC 0 = 2 D1 − 1.5D2 + D3 + 0 5 − = −1.5D1 + 1.5D2 − 1.5 D3 + 0 EI 0 = D1 − 1.5 D2 + 4 D3 + D4 0 = 0 + 0 + D3 + 2 D2 edayedaHRsayRbB½n§smIkarxagelI eyIgTTYl)an 16.67 D1 = − EI 26.67 D2 = − EI 6.67 D3 = − EI 3.33 D4 = EI edayeRbIlT§plTaMgenH ehIyedayKuNCYrQrTaMgBIrxageRkay eyIg)an ⎛ 16.67 ⎞ ⎛ 26.67 ⎞ ⎛ 3.33 ⎞ Q5 = 1.5 EI ⎜ − ⎟ − 1.5 EI ⎜ − ⎟ + 0 − 1.5 EI ⎜ ⎟ ⎝ EI ⎠ ⎝ EI ⎠ ⎝ EI ⎠ = 10kN ⎛ 6.67 ⎞ ⎛ 3.33 ⎞ Q6 = 0 + 0 + 1.5EI ⎜ − ⎟ + 1.5 EI ⎜ ⎟ ⎝ EI ⎠ ⎝ EI ⎠ = −5kN ]TahrN_ 15-2³ kMNt;kmøaMgkat; nigm:Um:g;enAkñúgGgát;elx ! rbs;FñwmEdlbgðajenAkñúgrUbTI 15-9a. EI CacMnYnefr. karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -510
  11. 11. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ kareFVIkMNt;smÁal;³ enAkñúgkrNIenH Fñwmman degree of freedom EdlCaGBaØatcMnUnEtBIrb:ueNÑaHEdlman bgðajelxkUd ! nig @ ¬rUbTI 15-9b¦. cMNaMfa bnÞúk M 0 manTMhMGviC¢man. m:aRTIsbnÞúk nigm:aRTIs bMlas;TIEdlCasmμtikmμKW ⎡0 ⎤ 3 ⎢0 ⎥ 4 ⎡ 0 ⎤1 Qk = ⎢ ⎥ Dk = ⎢ ⎥ ⎣− M 0 ⎦ 2 ⎢0 ⎥ 5 ⎢ ⎥ ⎣0 ⎦ 6 m:aRTIsPaBrwgRkajsRmab;Ggát;³ edayGnuvtþsmIkar 15-1 eTAelIGgát;nImYy² edayeyageTAelIelxkUd EdlbgðajenAkñúgrUbTI 15-9b eyIg)an 5 6 4 1 4 1 3 2 ⎡ 12 6 12 6 ⎤5 ⎡ 12 6 12 6 ⎤4 ⎢ 3 2 − 3 ⎥ ⎢ 3 − 3 ⎥ ⎢ L L L L2 ⎥ ⎢ L L2 L L2 ⎥ ⎢ 6 4 − 2 6 2 ⎥ 6 ⎢ 6 4 − 2 6 2 ⎥ 1 k1 = EI ⎢ L2 L L L ⎥ k 2 = EI ⎢ L2 L L L ⎥ ⎢ 12 6 12 6⎥ ⎢ 12 6 12 6⎥ ⎢− 3 − 2 − 2 ⎥4 ⎢− 3 − 2 − 2 ⎥3 ⎢ L L L3 L ⎥ ⎢ L L L3 L ⎥ ⎢ 6 2 − 2 6 4 ⎥ ⎢ 6 2 − 2 6 4 ⎥ ⎢ L2 ⎣ L L L ⎥1 ⎦ ⎢ L2 ⎣ L L L ⎥2 ⎦ bnÞúk nigbMlas;TI³ eKbegáItm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§edaypÁúMm:aRTIsPaBrwgRkajsRmab;Ggát; nImYy². edayGnuvtþsmIkarm:aRTIssRmab;rcnasm<½n§ eyIg)an Q = KD 1 2 3 4 5 6 ⎡ 8 2 −6 6 2 ⎤ ⎡ D1 ⎤ 1 ⎢ L 0 L L2 L2 L ⎥⎢ ⎥ ⎢ 2 4 6 6 ⎥⎢ ⎥ 1⎡ 0 ⎤ ⎢ − 2 0 0 ⎥ ⎢ D2 ⎥ 2 ⎢ L L2 ⎥⎢ ⎥ 2 ⎢− M 0 ⎥ ⎢ 6 L L ⎢ ⎥ 6 12 12 ⎥⎢ ⎥ 3 ⎢ Q3 ⎥ = ⎢− L2 − L2 L3 − L3 0 0 ⎥⎢ 0 ⎥ 3 ⎢ ⎥ ⎢ ⎥ 6 ⎥⎢ ⎥ 4 ⎢ Q4 ⎥ ⎢ 0 6 12 24 − 3 12 − 3 − 2 ⎢ 0 ⎥4 5 ⎢ Q5 ⎥ ⎢ L2 L L3 L L ⎥⎢ ⎥ ⎢ ⎥ ⎢ 6 12 12 6 ⎥⎢ ⎥ 6 ⎢ Q6 ⎥ ⎢ 2 ⎣ ⎦ 0 0 − 3 ⎥⎢ 0 ⎥ 5 ⎢ L L L3 L2 ⎥ ⎢ ⎥ ⎢ 2 0 0 − 2 6 6 4 ⎥⎢ ⎥ ⎢ L ⎣ L L2 L ⎥⎢ 0 ⎥ 6 ⎦⎣ ⎦ eFVIplKuNeTAelICYredkdMbUgTaMgBIredIm,IkMNt;bMlas;TI eyIg)an Beam analysis using the stiffness method T.Chhay -511
  12. 12. Department of Civil Engineering NPIC 8EI 2 EI 0= D1 + D2 L L 2 EI 4 EI − M0 = D1 + D2 L L dUenH M L D1 = 0 14 EI 2M 0 L D2 = − 7 EI dUcenAkñúg]TahrN_elIkmun eyIgTTYl)ankmøaMgRbtikmμBIkareFVIplKuNénCYredkEdlenAsl;. ]TahrN_ kmøaMgRbtikmμenAxagsþaMgédKW 6 EI ⎛ M 0 L ⎞ 6 EI ⎛ 2M 0 L ⎞ 9M 0 Q3 = − ⎜ ⎟− ⎜− ⎟= L2 ⎝ 14 EI ⎠ L2 ⎝ 7 EI ⎠ 7L eKkMNt;kmøaMgkñúgenARtg; node ! nig @ BIsmIkar 15-2. eyIg)an q = k1d 5 6 4 1 ⎡ q5 ⎤ ⎡ 12 6 12 6 ⎤⎡ 0 ⎤ 5 ⎢ ⎥ ⎢ 3 − 3 ⎥⎢ ⎥ ⎢ ⎥ ⎢ L L2 L L2 ⎥ ⎢ ⎥ ⎢q6 ⎥ ⎢ 6 4 6 2 ⎥⎢ 0 ⎥ − 2 6 ⎢ ⎥ = EI ⎢ L2 L L L ⎥⎢ ⎥ ⎢ ⎥ ⎢ 12 6 12 6 ⎥⎢ ⎥ ⎢q 4 ⎥ ⎢ − 3 − 2 − 2 ⎥⎢ 0 ⎥ 4 ⎢ ⎥ ⎢ L L L3 L ⎥⎢ ⎥ ⎢ ⎥ ⎢ 6 2 6 4 ⎥⎢ M L ⎥ − 2 L ⎥ ⎢14 EI ⎥ 1 0 ⎢ ⎥ ⎢ L2 ⎣ ⎦⎣ ⎣ q1 ⎦ L L ⎦ 6 EI ⎛ M L ⎞ 3M 0 q5 = 2 ⎜ 0 ⎟ = L ⎝ 14 EI ⎠ 7L 2 EI ⎛ M 0 L ⎞ M 0 q6 = ⎜ ⎟= L ⎝ 14 EI ⎠ 7 6 EI ⎛ M 0 L ⎞ 3M 0 q4 = − ⎜ ⎟=− L ⎝ 2 14 EI ⎠ 7L 4 EI ⎛ M 0 L ⎞ 2 M 0 q1 = ⎜ ⎟= L ⎝ 14 EI ⎠ L lT§plTaMgenHRtUv)anbgðajenAkñúgrUbTI 15-9c. ]TahrN_ 15-3³ FñwmenAkñúgrubTI 15-10a rgm:Um:g;BIr. RbsinebITMrkNþal ②Rsut 1.5mm cUgkMNt; kmøaMgRbtikmμTMrenARtg;TMr. yk E = 200GPa nig I = 22(10 −6 )m 4 . karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -512
  13. 13. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ kareFVIkMNt;smÁal;³ FñwmmanFatuBIr nig degree of freedom EdlCaGBaØatcMnYnbI. BYkvaRtUv)aneKkMNt; edayelxkUdtUcCageK ¬rUbTI 15-10b¦. enATIenH m:aRTIsbnÞúk nigm:aRTIsbMlas;TIEdlCasmμtikmμKW³ ⎡4 ⎤ 1 ⎡0 ⎤4 Qk = ⎢0 ⎥ 2 ⎢ ⎥ ⎢− 0.0015⎥ 5 Dk = ⎢ ⎥ ⎢ − 4⎥ 3 ⎣ ⎦ ⎢0 ⎣ ⎥6 ⎦ m:aRTIsPaBrwgRkajsRmab;Ggát;³ eKkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;edayeRbIsmIkar 15-1 eday eyageTAtamelxkUd nigTisrbs;Ggát;EdlbgðajenAkñúgrUbTI 15-10b. eyIg)an 6 3 5 2 5 2 4 1 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 6 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 5 ⎢ − 1.5 1 ⎥ 3 ⎢ − 1.5 1 ⎥ 2 k1 = EI ⎢ 1.5 2 ⎥ k 2 = EI ⎢ 1.5 2 ⎥ ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 5 ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 4 ⎢ ⎥ ⎢ ⎥ ⎣ 1.5 1 − 1.5 2 ⎦ 2 ⎣ 1.5 1 − 1.5 2 ⎦ 1 bMlas;TI nigbnÞúk³ karpÁúMm:aRTIsPaBrwgRkajsRmab;rcnasm<n§ nigkarsresrsmIkarPaBrwgRkajsRmab; ½ rcnasm<½n§ eyIg)an 1 2 3 4 5 6 ⎡ 4⎤ ⎡ 2 1 0 − 1.5 1.5 0 ⎤ ⎡ D1 ⎤ ⎢0⎥ ⎢ 1 4 1 − 1.5 0 1.5 ⎥ ⎢ D2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢− 4⎥ = EI ⎢ 0 1 2 0 − 1.5 1.5 ⎥ ⎢ D3 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q4 ⎥ ⎢− 1.5 − 1.5 0 1.5 − 1.5 0 ⎥ ⎢ 0 ⎥ ⎢Q5 ⎥ ⎢ 1.5 0 − 1.5 − 1.5 3 − 1.5⎥ ⎢− 0.0015⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎦ ⎣Q6 ⎥ ⎢ 0 ⎣ 1.5 1.5 0 − 1.5 1.5 ⎥ ⎢ 0 ⎦⎣ ⎥ ⎦ edaHRsayrkbMlas;TIEdlCaGBaØat = 2 D1 + D2 + 0 D3 − 1.5(0) + 1.5(− 0.0015) + 0 4 EI 0 = 1D1 + 4 D2 + 1D3 − 1.5(0 ) + 0 + 0 Beam analysis using the stiffness method T.Chhay -513
  14. 14. Department of Civil Engineering NPIC −4 = 0 D1 + 1D2 + 2 D3 + 0 − 1.5(− 0.0015) + 0 EI edayCMnYs EI = 200(10 6 )(22)(10 −6 ) nigedaHRsay eyIg)an D1 = 0.001580rad D2 = 0 D3 = −0.001580rad edayeRbIlT§plTaMgenH dUcenHkmøaMgRbtikmμTMrKW Q4 = 200( 6 )22( −6 )[− 1.5(0.001580 ) − 1.5(0 ) + 0 + 1.5(0 ) − 1.5(− 0.0015) + 0] = −0.525kN 10 10 Q5 = 200(10 6 )22(10 −6 )[1.5(0.001580 ) + 0 − 1.5(− 0.001580 ) − 1.5(0 ) + 3(− 0.0015) − 1.5(0 )] = 1.05kN Q6 = 200(10 6 )22( −6 )[0 + 1.5(0 ) + 1.5(− 0.001580 ) + 0 − 1.5(− 0.0015) + 1.5(0 )] = −0.525kN 10 ]TahrN_ 15-4³ kMNt;m:Um:g;EdlekItmanenARtg;TMr A rbs;FñwmEdlbgðajenAkñúgrUbTI 15-11a. yk E = 200GPa nig I = 216(10 6 )mm 4 . dMeNaHRsay³ kareFVIkMNt;smÁal;³ enATIenH Fñwmman degree of freedom EdlCaGBaØatcMnYnBI EdlkMNt;elxerogeday elxkUd ! nig @. karviPaKedaym:aRTIsTamTareGaybnÞúkGnuvtþenARtg; node dUcenHeKRtUvCMnYsbnÞúkBRgay nig bnÞúkcMcMNucedaym:Um:g;bgáb;cugsmmUlrbs;vaEdlRtUv)ankMNt;BItaragenAkñúgemeron !! ¬emIl]TahrN_ 11-2¦. cMNaMfa minmanbnÞúkxageRkAmanGMeBIenARtg; ① eT edaysarkmøaMgRbtikmμenARtg;elxkUd # karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -514
  15. 15. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa nig $ CaGBaØatenAkñúgm:aRTIsbnÞúk. edayeRbIviFItRmYtpl lT§plénkarviPaKm:aRTIssRmab;bnÞúkenAkñúgrUb TI 15-11b RtUv)anEktRmUvenAeBleRkayedaybnÞúkEdlbgðajenAkñúgrUbTI 15-11c. BIrUbTI 15-11b m:aRTIsbMlas;TI nigm:aRTIsbnÞúksmμtikmμKW ⎡0 ⎤ 4 ⎡12 ⎤ 1 Dk = ⎢0 ⎥ 5 ⎢ ⎥ Qk = ⎢ ⎥ ⎢0 ⎥ 6 ⎣84⎦ 2 ⎣ ⎦ m:aRTIsPaBrwgRkajsRmab;Ggát;³ eKkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;BIsmIkar 15-1. Ggát;elx !³ 12EI = 12(2003)(216) = 2400 3 L 6 6 EI 6(200 )(216) = = 7200 L2 62 4 EI 4(200 )(216 ) = = 28800 L 6 2 EI 2(200 )(216) = = 14400 L 6 4 3 5 2 ⎡ 2400 7200 − 2400 7200 ⎤ 4 ⎢ ⎥ k1 = ⎢ 7200 28800 − 7200 14400 ⎥ 3 ⎢− 2400 − 7200 2400 − 7200⎥ 5 ⎢ ⎥ ⎣ 7200 14400 − 7200 28800 ⎦ 2 12 EI 12(200 )(216 ) Ggát;elx @³ = = 64800 L3 23 6 EI 6(200 )(216) = = 64800 L2 22 4 EI 4(200 )(216) = = 86400 L 2 2 EI 2(200 )(216 ) = = 43200 L 2 5 2 6 1 ⎡ 64800 64800 − 64800 64800 ⎤ 5 ⎢ ⎥ k 2 = ⎢ 64800 86400 − 64800 43200 ⎥ 2 ⎢− 64800 − 64800 64800 − 64800⎥ 6 ⎢ ⎥ ⎣ 64800 43200 − 64800 86400 ⎦ 1 bMlas;TI nigbnÞúk³ eyIgtMrUveGay Q = KD Beam analysis using the stiffness method T.Chhay -515
  16. 16. Department of Civil Engineering NPIC 1 2 3 4 5 6 ⎡ 12 ⎤ ⎡ 86400 43200 0 0 64800 − 64800⎤ ⎡ D1 ⎤ ⎢ 84 ⎥ ⎢ 43200 115200 14400 7200 57600 − 64800⎥ ⎢ D ⎥ ⎢ ⎥ ⎢ ⎥⎢ 2 ⎥ ⎢Q3 ⎥ = ⎢ 0 14400 28800 7200 − 7200 0 ⎥⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q4 ⎥ ⎢ 0 7200 7200 2400 − 2400 − 64800⎥ ⎢ 0 ⎥ ⎢Q5 ⎥ ⎢ 64800 57600 − 7200 − 2400 67200 − 64800⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q6 ⎦ ⎢− 64800 − 64800 ⎣ ⎥ ⎣ 0 0 − 64800 64800 ⎥ ⎢ 0 ⎥ ⎦⎣ ⎦ edayedaHRsaytamrebobFmμtar 12 = 86400 D1 + 43200 D2 84 = 43200 D1 + 115200 D2 ( ) D1 = −0.2778 10 −3 m D2 = 0.8333( 10 )m −3 dUcenH Q3 = 0 + 14400(0.8333)(10 −3 )m = 12kN .m m:Um:g;Cak;EsþgenARtg; A RtUvEtrYmbBa©ÚlkmøaMgRbtikmμTMrbgáb; + 96kN.m EdlbgðajenAkñúgrUbTI 15-11c CamYynwglT§plEdl)anKNnasRmab; Q3 . dUcenH M AB = 12kN .,+96kN .m = 108kN .m lT§plenHdUcKñanwglT§plEdlKNnaenAkñúg]TahrN_ 11-2. eTaHbICaeKminRtUvkarKNnam:Um:g; nigkmøaMgkat;enARtg;cMNuc B enATIenHk¾eday EteKGacKitGac KNnava)an ]TahrN_ Ggát;elx !/ node @ ¬rUbTI 15-11b¦. lT§plTamTarkarBnøat q1 = k1d + (q0 )1 4 3 5 2 ⎡ q 4 ⎤ ⎡ 2400 7200 − 2400 7200 ⎤ ⎡ 0 ⎤ ⎡ 6 ⎤ ⎢ q ⎥ ⎢− 7200 28800 − 7200 14400 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 3⎥ = ⎢ ⎥⎢ ⎢ q5 ⎥ ⎢− 2400 − 7200 67200 57600 ⎥ ⎢ 0 ⎥ ( ) ⎥ 10 −3 + ⎢ 96 ⎥ ⎢ 6 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣q 2 ⎦ ⎣ 7200 14400 57600 115200⎦ ⎣0.833⎦ ⎣− 96⎦ ]TahrN_ 15-5³ kMNt;PaBdabenARtg;① nigkmøaMgRbtikmμTMrenAelIFñwmEdlbgðajenAkñúgrUbTI 15-12a. EI CacMnYnefr. karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -516
  17. 17. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ kareFVIkMNt;smÁal;³ FñwmRtUv)anEbgEckCaBIrFatu ehIy node nigGgát;RtUv)ankMNt;CamYynwgTisBIcug CiteTAcugq¶ay ¬rUbTI 15-12b¦. PaBdabEdlCaGBaØatRtUv)anbgðajenAkñúgrUbTI 15-12c. cMNaMfa bMlas;TImMu D4 minekItmaneT edayfakarTb;edayTMrrGilxag. m:aRTIsPaBrwgRkajsRmab;Ggát;³ edaysar EI CacMnYnefr ehIyGgát;manRbEvgesμIKña enaHm:aRTIsPaBrwg RkajsRmab;Ggát;RtUvEtdUcKña. edayeRbIelxkUdedIm,IkMNt;elxerogCYredk nigCYrQrnImYy²edayeyag eTAtamsmIkar 15-1 nigrUbTI 15-12b eyIg)an 3 4 1 2 1 2 5 6 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 3 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 1 ⎢ − 1.5 1 ⎥ 4 ⎢ − 1.5 1 ⎥ 2 k1 = EI ⎢ 1.5 2 ⎥ k 2 = EI ⎢ 1.5 2 ⎥ ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 1 ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 5 ⎢ ⎥ ⎢ ⎥ ⎣ 1.5 1 − 1.5 2 ⎦ 2 ⎣ 1.5 1 − 1.5 2 ⎦ 6 bMlas;TI nigbnÞúk³ pÁúMm:aRTIsPaBrwgRkajsRmab;Ggát;eTAkñúgm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§ eyIg)an Q = KD 1 2 3 4 5 6 ⎡− P ⎤ ⎡ 3 0 − 1.5 − 1.5 − 1.5 1.5 ⎤ ⎡ D1 ⎤ ⎢ 0 ⎥ ⎢ 0 4 1.5 1 − 1.5 1 ⎥ ⎢ D2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ 0 ⎥ = EI ⎢− 1.5 1.5 1.5 1.5 0 0 ⎥ ⎢ D3 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ Q4 ⎥ ⎢− 1.5 1 1.5 2 0 0 ⎥⎢ 0 ⎥ ⎢ Q5 ⎥ ⎢− 1.5 − 1.5 0 0 1.5 − 1.5⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ Q6 ⎥ ⎣ ⎦ ⎢ 1.5 ⎣ 1 0 0 − 1.5 2 ⎥ ⎢ 0 ⎥ ⎦⎣ ⎦ Beam analysis using the stiffness method T.Chhay -517
  18. 18. Department of Civil Engineering NPIC edayedaHRsayrkbMlas;TI eyIg)an P − = 3D1 + 0 D2 − 1.5D3 EI 0 = 0 D1 + 4 D2 + 1.5D3 0 = −1.5D1 + 1.5D2 + 1.5D3 1.667 P D1 = − EI P D2 = EI 2.667 P D3 = − EI cMNaMfa sBaØarbs;lT§plTaMgenHRtUvEtRtUvKñanwgTisedArbs;PaBdabEdlbgðajenAkñúgrUbTI 15-12c. kareRbIlT§plTaMgenH kmøaMgRbtikmμKW ⎛ 1.667 P ⎞ ⎛ P ⎞ ⎛ 2.667 P ⎞ Q4 = −1.5EI ⎜ − ⎟ + 1EI ⎜ ⎟ + 1.5EI ⎜ − ⎟ = −0.5P ⎝ EI ⎠ ⎝ EI ⎠ ⎝ EI ⎠ ⎛ 1.667 P ⎞ ⎛ P ⎞ ⎛ 2.667 P ⎞ Q5 = −1.5 EI ⎜ − ⎟ − 1.5EI ⎜ ⎟ + 0⎜ − ⎟=P ⎝ EI ⎠ ⎝ EI ⎠ ⎝ EI ⎠ ⎛ 1.667 P ⎞ ⎛ P ⎞ ⎛ 2.667 P ⎞ Q6 = 1.5EI ⎜ − ⎟ + 1EI ⎜ ⎟ + 0⎜ − ⎟ = −1.5P ⎝ EI ⎠ ⎝ EI ⎠ ⎝ EI ⎠ karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -518
  19. 19. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa cMeNaT !%>! kMNt;kmøaMgRbtikmμTMr. snμt;②CaTMrkl;. !%>% kMNt;m:Um:g;Bt;enAkñúgFñwmenARtg; ②nig③. EI CacMnYnefr. snμt;②CaTMrkl; ehIy③ nig④CaTMrsnøak;. EI CacMnYnefr. !%>@ kMNt;m:Um:g;Bt;enAkñúgFñwmenARtg; ① nig②. !%>^ kMNt;kmøaMgRbtikmμTMr. snμt;②CaTMrsnøak; snμt;②CaTMrkl; ehIy③CaTMrsnøak;. EI CacMnYn ehIy① nig③ CaTMrkl . EI CacMnYnefr. efr. !%>& kMNt;kmøaMgRbtikmμTMr. EI CacMnYnefr. !%># kMNt;kmøaMgRbtikmμTMr. EI CacMnYnefr. !%>* kMNt;m:Um:g;Bt;enARtg; ① nig③. snμt;② !%>$ kMNt;m:Um:g;Bt;enARtg;TMr. snμt;②CaTMrkl;. CaTMrkl; ehIy① nig③CaTMrbgáb;. EI CacMnYn EI CacMnYnefr. efr. Problems T.Chhay -519
  20. 20. Department of Civil Engineering NPIC !%>( kMNt;m:Um:g;Bt;enARtg; ① nig③ RbsinebI TMr②Rsut 30mm ¬cMeNaT !%>(¦. snμt; ②Ca TMrkl; ehIy① nig③CaTMrbgáb;. EI = 5000kN .m 2 . !%>!0 kMNt;kmøaMgRbtikmμTMr. EI CacMnYnefr. !%>!! kMNt;kmøaMgRbtikmμTMr. eKmanTMrrGiltam TisbBaÄrenARtg;① dUcbgðaj. EI CacMnYnefr. cMeNaT T.Chhay -520

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